Gunther Schauberger & Gerhard Tutz
BTLLasso - A Common Framework and Software
Package for the Inclusion and Selection of Covariates
in Bradley-Terry Models
Technical Report Number 202, 2017
Department of Statistics
University of Munich
http://www.stat.uni-muenchen.de
BTLLasso - A Common Framework and Software
Package for the Inclusion and Selection of
Covariates in Bradley-Terry Models
Gunther Schauberger
Gerhard Tutz
LMU Munich
LMU Munich
Abstract
In paired comparison models, the inclusion of covariates is a tool to account for the
heterogeneity of preferences and to investigate which characteristics determine the preferences. Although methods for the selection of variables have been proposed no coherent
framework that combines all possible types of covariates is available. There are three different types of covariates that can occur in paired comparisons, the covariates can either
vary over the subjects, the objects or both the subjects and the objects of the paired
comparisons. This paper gives an overview over all possible types of covariates in paired
comparisons and introduces a general framework to include covariate effects into BradleyTerry models. For each type of covariate, appropriate penalty terms that allow for sparser
models and therefore easier interpretation are proposed. The whole framework is implemented in the R-package BTLLasso. The main functionality and the visualization tools
of the package are introduced and illustrated by using real data sets.
Keywords: Bradley-Terry, paired comparison, penalization, variable selection, BTLLasso.
1. Introduction
Paired comparisons have a long tradition in statistics with the roots dating back to Zermelo
(1929). In many applications the objective is to find the underlying preference scale by presenting objects or items in pairs. The method has been used in various areas, for example, in
psychology, to measure the intensity or attractiveness of stimuli, in marketing, to evaluate the
attractiveness of brands, in social sciences, to investigate the value orientation, e.g. Francis,
Dittrich, Hatzinger, and Penn (2002). Paired comparison are also found in sports whenever
two players or teams compete in a competition. Then the non-observable scale to be found
refers to the strengths of the competitors.
The most widely used model for paired comparison data is the Bradley-Terry model. Originally proposed by Bradley and Terry (1952) it is also referred to as the Bradley-Terry-Luce
(BTL-) model indicating the strong connection to Luce’s choice axiom formulated in Luce
2
(1959). Luce’s choice axiom states that the decision between two objects is not influenced
by other objects, which is also known as the independence from irrelevant alternatives. Over
the years various extensions have been proposed allowing for dependencies among responses,
time dependence or simultaneous ranking with respect to more than one attribute. Overviews
were given by Bradley (1976), David (1988) and more recently by Cattelan (2012).
The main problem when applying the Bradley-Terry model is that it implies severe restrictions and assumptions. In particular it assumes that there is a one-dimensional latent scale
that represents the dominance, strength or attractiveness of objects. In experiments with
several persons or, more general, several subjects, the BTL-model is typically fitted under
the assumption that the strengths of the objects are equal for all subjects. Since the heterogeneity over subjects is ignored the model often yields a bad fit. The assumption of a
latent scale becomes much weaker if one assumes that the strengths of objects depends on
covariates. The covariates can be subject-specific or object-specific. The former refers to the
characteristics of the subjects who choose between alternatives/objects, the latter refers to
the characteristics of the objects which are compared. Also subject-object-specific covariates,
that is, characteristics that vary both over subjects and over objects, are possible. Explicit
modeling of heterogeneity is rare, and usually restricted to few covariates, see Turner and
Firth (2012) or Francis, Dittrich, and Hatzinger (2010). Software packages for the analysis
of paired comparisons that are available in R packages (R Core Team 2016), the software
environment we also use, are the package BradleyTerry2 (Turner and Firth 2012) and prefmod (Hatzinger and Dittrich 2012). However, they do not contain built-in procedures for
the selection of variables. More recently, methods that are able to handle a larger number
of explanatory variables have been proposed, Casalicchio, Tutz, and Schauberger (2015) presented a boosting approach and developed the corresponding R-package ordBTL (Casalicchio
2013). Strobl, Wickelmaier, and Zeileis (2011) used recursive partitioning techniques and
implemented partitioned paired comparison models in the R-package psychotree.
The objective of the present paper is to develop a common framework for the inclusion and
selection of covariates in BTL-models based on regularization techniques. The inclusion of
covariates serves two purposes. It accounts for heterogeneity in the population yielding more
realistic models and it allows to investigate which variables determine the strengths of objects.
The proposed methods are made available in the program package BTLLasso, the use of the
package is described and illustrated.
2. The Bradley-Terry model
In the following the simple binary Bradley-Terry model and extended versions that allow for
ordinal responses are briefly sketched.
2.1. The basic model
Assuming a set of objects ta1 , . . . , am u, in its most simple form the (binary) Bradley-Terry
model is given by
P pa r
p γr γs q .
¡ asq P pYpr,sq 1q 1 expexp
pγ γ q
r
(1)
s
The response of the model represents the probability that a certain object ar dominates (or is
3
preferred over) another object as , ar ¡ as . The response is formalized in the random variable
Ypr,sq which is defined to be Ypr,sq 1 if ar is preferred over as and Ypr,sq 0 otherwise.
The parameters γr , r 1, . . . , m, represent the attractiveness or strength of the respective
objects. For identifiability, a restriction on the parameters is needed, for example m
r1 γr 0
or γm 0.
°
2.2.
Bradley–Terry models with ordered response categories
In many applications the dominance of one of the objects is quite naturally observed on an
ordered scale. For example, in sport competitions one often has to account for draws. Early
extensions of the BTL-model include at least the possibility of ties, see Rao and Kupper
(1967), Glenn and David (1960) and Davidson (1970). General models for ordered responses,
for example to allow for a 5-point scale (much better, slightly better, equal, slightly worse,
much worse) were proposed by Tutz (1986) and Agresti (1992). For K response categories,
the model parametrizes the cumulative probabilities
P pYpr,sq
pθ k
¤ kq 1 expexp
pθ
k
γr γs q
γr γs q
(2)
with k 1, . . . , K denoting the the possible response categories. The parameters θk represent
the so-called threshold parameters for the single response categories, they determine the
preference for specific categories. In particular, Ypr,sq 1 represents the maximal preference
for object ar over as and Ypr,sq K represents the maximal preference for object as over ar .
In general, for ordinal paired comparisons it can be assumed that the response categories have
a symmetric interpretation so that P pYpr,sq k q P pYps,rq K k 1q holds. Therefore,
the threshold parameters should be restricted with θk θK k and, if K is even, θK {2 0 to
guarantee for symmetric probabilities. The threshold for the last category is fixed to θK 8
so that P pYpr,sq ¤ K q 1 will hold. The probability for a single response category can
be derived from the difference between two adjacent categories, P pYpr,sq k q P pYpr,sq ¤
k q P pYpr,sq ¤ k 1q. To guarantee for non-negative probabilities for the single response
categories one restricts θ1 ¤ θ2 ¤ . . . ¤ θK . The ordinal Bradley-Terry model corresponds to
a cumulative logit model and can be estimated using methods from this general framework.
2.3. Bradley-Terry models with order effects
In some specific paired comparisons it can be decisive in which order the competing objects
are presented. Typical examples are sports events as, for example, matches in the German
Bundesliga as considered in an exemplary data set in this article. Here, the first-mentioned
team typically is the team playing at its home ground where it might have a (home) advantage
over its opponent. Therefore, the assumption that the response categories are symmetric does
not hold anymore and the model needs to be adapted accordingly. Extending the basic models
by an additional parameter δ yields the binary Bradley-Terry model
P pYpr,sq
and the ordinal model
P pYpr,sq
pδ γ r γ s q
1q 1 expexp
pδ γ γ q
p δ θk
¤ kq 1 expexp
pδ θ
r
k
s
γr γs q
.
γr γs q
4
Here, δ denotes the order effect which is simply incorporated into the design matrix by an
additional intercept column. If δ ¡ 0, it increases the probability of the first-named object
ar to win the comparison or, in the case of an ordinal response, to achieve a superiour result.
Given the order effect, the symmetry assumption for the response categories still holds.
3. The general Bradley-Terry model with explanatory variables
In many applications covariates that characterize the subjects or/and the objects are available. These can be used to model the heterogeneity in the population and investigate, which
variables determine the choice between alternatives. We first describe two applications that
will be used later, the respective data sets can be found in the package BTLLasso.
3.1. Exemplary data sets
Election data
The German Longitudinal Election Study (GLES) is a long-term study of the German electoral
process, see Rattinger, Roßteutscher, Schmitt-Beck, Weßels, and Wolf (2014). It collects preand post-election data for several federal elections. The specific data we are using here
originate from the pre-election survey for the German federal election in 2013. In this specific
part of the study, 2003 persons were asked to rate the most important parties (CDU/CSU,
SPD, Greens, Left Party, FDP, we eliminated the smaller parties AfD and the Pirate Party)
for the upcoming federal election on a scale from 5 to 5. The rating scales Zir reflect
the general opinions of participant i on party r with 5 representing a very positive and
5 representing a very negative opinion. The original scales were transformed into paired
comparisons by building all pairwise differences of the scores of all five parties per person.
Using this procedure one ends up with ordered paired comparisons with values between 10
and 10. The response was narrowed down to an ordered response with five categories. The
data now represent paired comparisons between all parties measured on an ordered five-point
scale:
Zir Zis
P t6, 10u
Zir Zis P t1, 5u
Zir Zis 0
Zir Zis P t5, 1u
Zir Zis P t10, 6u
ÞÑ
ÞÑ
Þ
Ñ
ÞÑ
ÞÑ
1:
Yipr,sq 2 :
Yipr,sq 3 :
Yipr,sq 4 :
Yipr,sq 5 :
Yipr,sq
”Strong preference of party r over party s”
”Weak preference of party r over party s”
”Equal opinions on parties r and s”
”Weak preference of party s over party r”
”Strong preference of party s over party r”
Here, a first general difference in contrast to the paired comparison models from Section 2
can be seen. Besides the compared objects ar and as , the response Yipr,sq now also depends
on the subject i that executes the comparison between both objects.
Two different types of covariates are available in the GLES data set, namely subject-specific
covariates and subject-object-specific covariates. While the first type of covariates only characterizes the participants of the study, the second type of covariates characterizes both the
participants and the parties. A more detailed distinction between the possible types of covariates in paired comparisons follows in Section 3.2. For the sake of simplicity, we restrict
5
to the following three covariates (out of eight covariates in the data set GLES in the package)
that characterize the participants of the study:
ˆ Age: age of participant in years
ˆ Gender : female (1); male (0)
ˆ Abitur School leaving certificate (1: Abitur/A levels; 0: else)
The main question regarding these variables will be, if and which covariates influence the preference for single parties. In addition to these person-specific covariates also three covariates
are available that vary both over the persons and over the parties. These variables measure
the (self-perceived) distances of participants to the single parties with respect to various political topics. These so-called political dimensions refer to the socio-economic dimension and
the topics of immigration and climate change. The participants are supposed to report both
their self-perception and their perception of the single parties toward a topic on a Likert scale
with 11 ordered levels. For example, for the topic of climate change the Likert scale offers a
continuum between the following statements:
Level 1: Fight against climate change should take precedence, even if it impairs economic growth.
Level 11: Economic growth should take precedence, even if it impairs the fight against climate
change.
The absolute difference between these perceptions can be seen as the (self-perceived) absolute
distance between the person (subject) and the parties (objects) with respect to a political
topic. Therefore, the variables SocEc, Immigration and Climate are available separately for
all five parties.
Football data
The second data set we consider are data from the season 2015/2016 of the German Bundesliga. The German Bundesliga is played as a double round robin between 18 teams. Table 1
shows the final table of the season 2015/16 together with the abbreviations of the teams used
in the application later. As in all three previous seasons, Bayern München won the championship. VfB Stuttgart and Hannover 96 were relegated to the second division.
In the data set, one match is treated as a paired comparison of both teams with respect
to their playing abilities. In total, a Bundesliga season has 34 matchdays with 9 matches
per matchday. Therefore, in total 306 matches are observed. The response variables Yipr,sq
represent the outcome of a match between team ar (as the home team) and team as on
matchday i. We use a 5-point scale defined by
$
'
1
'
'
&2
Yipr,sq 3
'
'
4
'
%5
if
if
if
if
if
team ar wins by at least 2 goals difference
team ar wins by 1 goal difference
the match ends with a draw
team as wins by 1 goal difference
team as wins by at least 2 goals difference.
6
Position
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
BAY
DOR
LEV
MGB
S04
MAI
BER
WOB
KOE
HSV
ING
AUG
BRE
DAR
HOF
FRA
STU
HAN
Team
Goals For
Goals Against
Points
Bayern München
Borussia Dortmund
Bayer 04 Leverkusen
Bor. Mönchengladbach
FC Schalke 04
1. FSV Mainz 05
Hertha BSC
VfL Wolfsburg
1. FC Köln
Hamburger SV
FC Ingolstadt 04
FC Augsburg
Werder Bremen
SV Darmstadt 98
TSG Hoffenheim
Eintracht Frankfurt
VfB Stuttgart
Hannover 96
80
82
56
67
51
46
42
47
38
40
33
42
50
38
39
34
50
31
17
34
40
50
49
42
42
49
42
46
42
52
65
53
54
52
75
62
88
78
60
55
52
50
50
45
43
41
40
38
38
38
37
36
33
25
Table 1: Final table of the German Bundesliga in the season 2015/2016.
In addition to the match results encoded by ordinal paired comparisons several covariates are
given. In particular, specific performance measurements of the single teams in single matches
are available. For example, for every team it is known what distance the team run in a certain
match or the percentage of ball possession per team. In the German Bundesliga, the data
supplier opta (http://www.optasports.com/) provides the data. The data we use are freely
available from the website of the German football magazine kicker (http://www.kicker.de/).
Again, only two covariates (out of seven covariates in the data set Buli1516 in the package)
are considered (per team and per match):
ˆ Distance: Total amount of km run
ˆ BallPossession: Percentage of ball possession
The research question here is to find whether Distance and BallPossession are decisive variables for the strength of a team and if these effects differ across teams. Beside these two
variables, also the average market values of the players of the single teams are known. The
market values are constant across the whole season and, in contrast to the previous variables,
are not specific to single matchdays but only to single teams. Therefore, distance and ball
possession are subject-object-specific covariates while market values are object-specific.
3.2. Model specification
In general, when modelling paired comparison data three different types of covariates have
to be distinguished. All three types of covariates occur in the exemplary data presented in
7
Section 3.1 and will be described in detail in the following. However, first a clear distinction
between subjects and objects in paired comparisons has to be made. In our denotation objects
refer to the units that are compared in a paired comparison experiment. For example, in the
GLES data from Subsection 3.1.1, the parties are the objects which vary in their attractiveness
to voters. In football matches (as considered in the data from Section 3.1.2), the teams are
the objects that are compared to measure their playing abilities. In contrast, the subjects in
paired comparisons refer to the units that actively make a decision concerning the dominance
of objects in a paired comparisons. In the election data example the voters (or rather the
participants of the study) represent the subjects. In football, the definition of subjects is
less obvious but also here subjects can be identified. We consider the day of the match
or the match itself, which can be characterized by covariates, as the subject of the paired
comparisons.
In the following, we distinguish between subject-specific covariates xi , object-specific covariates z r and subject-object-specific covariates z ir . In the general framework proposed
here, those covariates can be included (simultaneously) into Bradley-Terry models. For
that purpose we consider the (ordinal) Bradley-Terry model in a generalized form. Let
Yipr,sq P 1, . . . , K denote the response for subject i if the objects par , as q are presented. The
ordering in the tuple par , as q is not arbitrary, it may represent the ordering of presentation
(ar first, as second) or the location of the meeting of teams (ar at home, as not at home).
Then the model has the form
P pYipr,sq
¤ kq 1
exppηiprsqk q
exppηiprsqk q
p δ θk
1 expexp
pδ θ
k
γir γis q
γir γis q
(3)
or equivalently
log
P pYipr,sq
P pYipr,sq
¤ kq
¡ kq δ
γir γis ,
θk
k
1, . . . , K 1.
(4)
In the model, the global object parameters γr are replaced by subject-specific parameters γir
for subjects i, i 1, . . . , n and objects ar , r 1, . . . , m. The extension allows to model
heterogeneity of subjects by including object-specific and subject-(object-)specific covariates.
Generally, for all possible covariates we assume a linear dependence structure with the linear
predictors of the model given by
ηiprsqk
δ
θk
γir γis .
Subject-specific covariates
Subject-specific covariates characterize the subjects that perform the comparisons between
the objects. In the case of the election data, the participants of the study represent the
subjects. Covariates like age, gender or unemployment status of the participants are treated
as subject-specific covariates, which may modify the attractiveness of parties. This is obtained
by assuming that the attractiveness of the parties (the strength parameters γir ) have the form
γir
βr0
xTi β r ,
where xi is a vector of subject-specific covariates. The inclusion of subject-specific covariates
into a paired comparison model automatically yields object-specific effects β r , which vary over
8
objects or alternatives. In the election study, for example, one wants to know if and how the
attractiveness of parties depends on the gender of the voters. Consequently, for every party
a distinct (object-specific) gender effect has to be specified. Accordingly, the attractiveness
of object ar depends on covariates xi of subject i. The parameter βrj represents the increase
of attractiveness of object r if the jth variable increases by one unit. To obtain identifiability
of the parameters, symmetric side constraints are used, i.e. one postulates m
r1 βrj 0 for
all j.
°
The inclusion of subject-specific covariates strongly weakens the assumptions on the model. It
is no longer assumed that the strengths of the objects are fixed and equal for all subjetcs when
they choose between a pair of objects. The model explicitly accounts for heterogeneity over
subjects. Subject-specific covariates in paired comparisons were considered before by Turner
and Firth (2012), Francis et al. (2002) or Francis et al. (2010). More recently, Casalicchio
et al. (2015) presented a boosting approach and Schauberger and Tutz (2015) a penalization
approach based on an L1 -penalty, both are able to include explanatory variables and select
the relevant ones. An alternative approach has been proposed by Strobl et al. (2011). It is
based on recursive partitioning techniques (also known as trees) and automatically selects the
relevant variables among a potentially large set of variables.
Subject-object-specific covariates
Covariates may also vary both over subjects and objects yielding subject-object-specific covariates. For example, in election studies as in the election data presented above it is frequently of
interest how political issues determine the preference of specific parties. The so-called spatial
election theory assumes that each party is characterized by a position in a finite-dimensional
space, with each dimension corresponding to one political issue. Spatial election theory then
explains the party choice of voters by a utility function that depends on the distance between
the voter’s own position and that of the parties within the space of policy-dimensions, for
details see, for example, Thurner and Eymann (2000) and Mauerer, Pössnecker, Thurner, and
Tutz (2015). In the election data, the covariates SocEc, Immigration and Climate measure
the distance between participants and the single parties on three different political dimensions
and are, therefore, considered as subject-object-specific covariates.
The effect of subject-object-specific covariates can be modelled in the same way as the effect
of subject-specific covariates. With z ir denoting the corresponding covariate vector one uses
γir
βr0
z Tir αr .
We use the notation z ir for the vector and αr for the weights to distinguish the cases subjectspecific and subject-object-specific covariates.
In contrast to subject-specific covariates, subject-object-specific covariates can also be modeled with global effects, which yields the simpler form
γir
βr0
z Tir τ .
The underlying assumption is that the effect strength is the same over all categories. That
means, for example, that the distance between the respondents parties position on an issues
like climate change have the same relevance for all parties. With a constraint on the intercepts,
for example, m
r1 βr0 0, parameters are identified if there is enough variation over subjects.
°
9
It may be checked by investigating if the corresponding design matrix for all pair comparisons
has full rank.
Object-specific covariates
It is more difficult to include pure object-specific covariates. Object-specific covariates characterize the compared objects but are constant across the subjects. In the football data, the
market value is an example for such an object-specific variable because the market values
vary across teams but are constant across matchdays. Similarily to the procedure proposed
here, Tutz and Schauberger (2014) included an object-specific covariate in an analysis on the
German Bundesliga.
The contribution of the object-specific covariates z r to the strength of object ar may be
specified by
γir
γr βr0
z Tr τ ,
where τ is a global parameter. Then z Tr τ represents the part of the attractiveness that is due
to the covariates and βr0 represents the part that is not explained by covariates. For example,
in the election data if one is interested if governing makes parties attractive one can use a
dummy variable z 1 for parties in the government and z 0 for parties of the opposition.
Then τ represents the importance of being part of the government for the attractiveness of a
party. In the case of object-specific covariates, no heterogeneity across subjects is modelled.
It is obvious that the parametrization leads to identifiability problems and additional side
constraints are needed. Let us consider the simple example of a binary covariate, as before
let z 1 indicate that the party is part of the government and z 0 indicate that the party
is not part of the government. Then the parties are nested within the factor z. Additional
side constraints have to restrict the parameters within the blocks formed by z 1 and z 0.
One may use j PM1 βj0 0, j PM0 βj0 0, where M1 , M0 is a partition of the parties with
M1 (M0 ) collecting all the parties with z 1 (z 0), respectively. Identifiability issues will
be considered in more detail in the following sections.
°
°
Intercepts
In most circumstances, in addition to the different covariate effects also object-specific intercepts βr0 are needed in the model. Without covariates, the intercepts correspond to the
regular strength/ability parameters from the basic Bradley-Terry models (1) or (2). Also
for the intercept parameters the additional side constraint m
r1 βr0 0 is necessary for
identifiability.
°
Order effects
As already mentioned in Section 2 in some circumstances the order of the competing objects
can be decisive and order effects are necessary. Besides a global order effect δ which is
equal across all objects also object-specific order effects δr are possible. For example, in the
application to the data from the German Bundesliga an order effect is clearly necessary as
based on experience a team playing at its home ground has an advantage over the away team.
If object-specific order effects are considered, one allows for different home effects across the
teams.
10
To sum up all options how to specify components in the general paired comparison model
(3), Table 2 contains all types of covariates and all possible parameterizations we consider.
ηiprsq
Effect type
γir
intercept
object-spec.
βr0
βs0
subject-spec. xi
object-spec.
xi β r
xi β s
subject-object-spec. z ir
global
z Tir τ
z Tis τ
global
z Tr τ
z Ts τ
pzir zis q τ
pz r z s q τ
subject-object-spec. z ir
object-spec.
z Tir αr
z Tis αs
z Tir αr - z Tis αs
order effect
global
δ
δ
order effect
object-spec.
δr
δr
ë incl. object-spec. zr
γis
Covariate type
T
T
βr0 βs0
xTi pβ r β s q
T
T
Table 2: Overview over parameterizations of covariate effects available in BTLLasso
3.3. Representation as generalized linear model
It is straightforward to embed the considered models into the framework of generalized linear
models (GLMs). The basic model (4) is a so-called cumulative ordinal regression model, which
is a multivariate GLM. It has the form logpP pYipr,sq ¤ k q{P pYipr,sq ¡ k qq ηipr,sqk with linear
predictors
ηipr,sqk δ θk γir γis , k 1, . . . , K 1.
All the models considered in the previous section are also multivariate GLMs with more structured predictors. For example, in the case of subject-object-specific covariates the predictor
has the form
ηipr,sqk
or
ηipr,sqk
δ
δ
θk
θk
βr0 βs0
βr0 βs0
pzir zisq
τ,
k
1, . . . , K 1
z Tir αr z Tis αs ,
k
1, . . . , K 1.
T
As far as estimation by maximum likelihood is concerned the only difference is in the structure
of the linear predictor and the design matrix, which does not only consist of dummies for the
objects as in the basic model.
Thus, in principle all the tools provided by GLMs, including tests, computation of standard
errors, and algorithms for the maximization of the log-likelihood, are available. For the theory
and computational tools, see, for example, Fahrmeir and Tutz (1997) or Tutz (2012).
4. Regularized estimation
In BTL models the inclusion of covariates yields models that contain a huge number of
parameters. If one includes, for example, a p-dimensional vector of subject-specific covariates
one has to estimate an pp 1q-dimensional parameter vector pβr0 , β r q for each object, which
already for moderate number of objects and size of the covariate vector yields a large number
of parameters to estimate. Therefore, in applications ML estimates tend to be unstable or
may not even exist.
11
One of the main features of BTLLasso is the selection of relevant terms and the implicit
reduction of the complexity of the model. In the simplest case selection refers to the selection
of variables, since typically in high-dimensional settings not all of the explanatory variables
have an impact on the preference of objects. However, model complexity can also be reduced
by identifying clusters of objects that share the same strength or variable effects.
For the selection of relevant terms, we propose to use a penalized likelihood approach. Instead
of maximizing the general log-likelihood lpξ q, one maximizes the penalized log-likelihood
lp pξ q lpξ q λJ pξ q
where lpξ q is the usual log-likelihood with ξ denoting a vector that contains all the parameters
of the model. J pξ q is a penalty term that penalizes specific structures in the parameter vector.
The parameter λ is a tuning parameter that specifies how seriously the penalty term has to
be taken. For the extreme choice λ 0 one obtains the ML estimate. The choice of the
tuning parameter is discussed later.
4.1. Penalties
Whatever the type of the variables and the components included in the model, the threshold
parameters θk of the ordinal model will never be penalized. For the other model components
we propose fusion and selection penalties, which are given in the following.
Clustering of Intercept Parameters
The penalty
P1 pβ10 , . . . , βm0 q ¸
|βr0 βs0|
r s
aims at the identification of clusters of objects that share the same intercept parameter. For
λ Ñ 8 one obtains one big cluster with β10 βm0 . Due to the symmetric side constraint
that means that β10 βm0 0 and all intercept parameters are eliminated from the
model. For appropriately chosen tuning parameter λ ideally the clusters of objects that have
the same intercepts are found. If no covariates are included the intercepts are equivalent to
the total strength of the objects. Therefore, the penalty is a tool to identify the objects that
really have to be distinguished. If explanatory variables are included the interpretation of
the clusters refers to the strength or attractiveness left after accounting for the effects of the
variables.
For pure object-specific covariates the attractiveness has the form γir βr0 z Tr τ . In this
case, the penalty provides a way to obtain unique parameter estimates despite the nonidentifiability of the model. The penalty serves to find out how much of the attractiveness
of an object is due to observed characteristics of the objects. When used in addition to the
regular side constraint m
r1 βrj 0 it restricts the intercepts and, therefore, the individual
attractiveness that is left if one accounts for the object-specific covariates. If the tuning
parameter gets large, λ Ñ 8, all strength parameters βr0 are estimated as identical and
the total strength is determined solely by z Tr τ . The corresponding model assumes that the
explanatory variables totally determine the abilities. This restricted model was proposed by
Springall (1973), however, it is hardly appropriate when a limited number of explanatory
variables is available.
°
12
Clustering and selection of variable effects
For object-specific effects, which are used for subject-specific covariates, the penalty
P 2 pβ 1 , . . . , β m q p ¸
¸
x
|βrj βsj |
j 1r s
yields for all variables (denoted by j) clusters of objects that share the same effect. Let
us consider the variable gender in the preference for parties. If βrj is the same for r P R,
comparisons between parties from a set of parties R do not depend on gender. However,
p1q
p2q
if βrj β.j for r P Rp1q and βrj β.j for r P Rp2q , gender distinguishes between the
groups of parties Rp1q and Rp2q . As in the case of the intercept parameters, side constraints
like m
r1 βrj 0 for every covariate j are needed. If λ increases successively all effects
corresponding to single covariates are merged to one cluster and the corresponding covariates
are eliminated from the model as all effects become zero. For finite λ some of the covariates
are deleted and for the others clusters of objects that share the same effect are identified.
°
For global effects, which are used for object-specific or subject-object-specific covariates, one
uses the penalty
P3 p τ q
p
¸
2
|τj |.
j 1
It is a simple lasso penalty that restricts the absolute values of the parameters. For λ
all corresponding covariates are eliminated from the model.
Ñ8
For object-specific parameters that are used on subject-object-specific covariates, we use the
modified penalty
P4 pα1 , . . . , αm q
p ¸
¸
1
|αrj αsj |
p
¸
1
ν1
j 1r s
m̧
|αrj |,
ν1
P t0, 1u .
j 1r 1
For this kind of parametrization, no further side constraints are needed. Therefore, the
respective penalty is composed of two parts. The first part penalizes all absolute pairwise
differences between the coefficients corresponding to one covariate, the second part (optionally
for ν1 1) penalizes the absolute values of all parameters. By specifying ν1 , the user has the
possibility to decide if only clustering of the objects with regard to the covariates is intended
(ν1 0) or if, additionally, selection of the covariates should be enabled (ν1 1). For λ Ñ 8,
the specification of ν1 leads to different results. While for ν1 0 one ends up with global
effects (equal for all objects but not zero), for ν1 1 all respective covariates are eliminated
from the model.
Clustering and selection of order effects
The order effects are penalized by the terms
P5 pδ q
or
P6 pδ1 , . . . , δm q
¸
r s
|δ |
|δr δs|
m̧
ν2
r 1
|δr |,
ν2
P t0, 1u,
13
depending on whether a global order effect δ is specified or an object-specific order effect
δr . The first penalty P5 is used if one starts with one order effect that does not vary over
objects, and only serves the purpose to restrict the size of the effect, which, in particular,
may be zero. Penalty P6 for object-specific order effects allows to identify clusters of objects
that share the same order effect. Additionally, the parameter ν2 allows for a decision between
solely clustering the order effects (ν2 0) or additional selection of the order effects (ν2 1).
With growing tuning parameter λ, for ν2 0 one ends up with one global order effect while
for ν2 1 all order effects can be eliminated from the model.
4.2. Combining penalties
Table 3 shows an overview on all presented penalty terms together with the corresponding
type of covariate.
Covariate type
Effect type
Penalty
intercept
object-spec.
P1 pβ10 , . . . , βm0 q subject-spec. xi
subject-object-spec. z ir
ë incl. object-spec. zr
subject-object-spec. z ir
object-spec.
P2 p β 1 , . . . , β m
global
3
p2
global
order effect
object-spec.
s0
rj
sj
j 1r s
j
j
P4 p α 1 , . . . , α m q P5 pδ q |δ |
order effect
r0
j 1
p2
j 1
object-spec.
r s
px
° |τ |
°
|τ |
P pτ q P3 p τ q global
° |β β |
°
q ° |β β |
P6 p δ 1 , . . . , δ m q ° ° |α α |
p1
rj
sj
j 1r s
° |δ δ |
r
r s
s
° ° |α |
p1
ν1
m
rj
j 1r 1
° |δ |
m
ν2
r
r 1
Table 3: Overview over all penalty terms on covariate effects available in BTLLasso
All penalty terms can use internal weights according to the principle of adaptive lasso (Zou
2006) using the option adaptive = TRUE from the function ctrl.BTLLasso(). For the sake
of simplicity the adaptive weights have been omitted in the presentation of the penalty terms.
Finally, all different penalty terms are combined using
J pξ q Ļ
ψl Pl ,
l 1
where ψl are penalty-specific weights. Although different tuning parameters for the different
penalty terms are conceivable, the optimization of tuning parameters would become overly
complicated. Therefore, all penalty terms are combined into one joint penalty controlled by
the joint tuning parameter λ. In order to allow for comparability of the different penalty
terms, two requirements have to be fulfilled. First, all covariates have to be transformed into
comparable scales. For that purpose,
ˆ the subject-specific covariates are scaled so that each vector x.1 , . . . , x.px referring to
one covariate across all subjects has variance 1.
14
ˆ the subject-object-specific covariates are scaled so that each vector z ..1 , . . . , z ..p1 referring to one covariate across all subjects and all objects has variance 1.
ˆ the object-specific covariates are scaled so that each covariate vector z .1 , . . . , z .p2 referring to one covariate across all objects has variance 1.
By default, the scaling listed above is executed automatically if scale = TRUE in ctrl.BTLLasso().
It should be noted that all print() commands and all visualization tools include the option
rescale to specify whether the estimates should be rescaled to the original scale or not.
While the rescaled estimates can be interpreted directly with respect to the corresponding
covariates, only the scaled parameters can be compared to each other in terms of effect sizes.
Second, the different penalty terms have to be weighted with weights ψl according to the
number of penalties and according to the number of free parameters they include, following
the principle of weighting penalties proposed by Bondell and Reich (2009) and Oelker and
Tutz (2015). The weights are applied if weight.penalties = TRUE in ctrl.BTLLasso() and
are multiplied by the weights (possibly) defined by adaptive = TRUE. The optimal tuning
parameter is chosen by k-fold cross-validation which is described in more detail in the following
section.
4.3. Implementation
In BTLLasso, penalized Fisher scoring is used to fit the proposed models. Generally, all
implemented penalties listed in Table 3 are L1 penalties and, therefore, are not differentiable.
However, following the suggestions of Fan and Li (2001) and Oelker and Tutz (2015) L1
penalties can be approximated by quadratic terms. Quadratic penalty terms are differentiable
and, therefore, can easily be incorporated in a (penalized) Fisher scoring algorithm. A first
implementation of quadratically approximated L1 penalties can be found in the R-package
gvcm.cat (Oelker 2015), but the package does not contain an implementation for cumulative
logit models. For shorter computation time, the fitting algorithm itself is implemented in C++
and integrated into R using the packages Rcpp (Eddelbuettel, François, Allaire, Chambers,
Bates, and Ushey 2011; Eddelbuettel 2013) and RcppArmadillo (Eddelbuettel and Sanderson
2014).
5. Use of the program package
The two main functions of the package BTLLasso are the functions
BTLLasso(Y, X = NULL, Z1 = NULL, Z2 = NULL, lambda,
control = ctrl.BTLLasso(), trace = TRUE)
and
cv.BTLLasso(Y, X = NULL, Z1 = NULL, Z2 = NULL, folds = 10, lambda,
control = ctrl.BTLLasso(), cores = folds, trace = TRUE,
trace.cv = TRUE, cv.crit = c("RPS", "Deviance"))
15
which carry out the penalized estimation of the specified model for a pre-specified grid of
tuning parameters λ. Compared to BTLLasso(), cv.BTLLasso() additionally performs crossvalidation to find the optimal value for the tuning parameter. The function ctrl.BTLLasso()
contains many different options to control the model fit.
Of course, also simple Bradley-Terry models without covariates as described in Section 2
can be computed with BTLLasso. Therefore, BTLLasso can also be a useful tool for the
computation of Bradley-Terry models with ordinal response or for models with (possibly
object-specific) order effects. For that purpose, all arguments to supply covariates (X, Z1 and
Z2) and all penalties can be ignored.
5.1. Model specification
In the following, it is generally described which functions and arguments are needed to specify
a paired comparison model with BTLLasso. More details and illustrations can be found in
Section 6 where specific examples based on exemplary data sets are presented.
Response
Both in BTLLasso() and cv.BTLLasso(), the response (i.e. the paired comparisons) is
specified using the argument Y. One needs to create a response object using the function
response.BTLLasso(), where one needs to declare the arguments response, first.object,
second.object and (possibly) subject. Only if a subject is specified, the paired comparisons
can be related to subject-specific or subject-object-specific covariates.
Covariates
In the functions BTLLasso() and cv.BTLLasso(), three different arguments are provided
to specify all covariates that should be included into a model, namely the arguments X, Z1
and Z2. The argument X contains all subject-specific covariates xi , the number of subjectspecific covariates (the number of columns of X) is denoted by px . In contrast, Z1 and Z2
contain object-specific covariates. In particular, Z2 contains all object-specific covariates zr
or subject-object-specific covariates zir which are modeled with global parameters τ . In total,
Z2 contains p2 covariates. Z1 contains p1 subject-object-specific covariates zir modeled with
object-specific parameters αr . Table 4 contains an overview over all possible assignments of
covariates.
Argument
Covariate type
Effect type
Parameter
Penalty
X
subject-spec. xi
object-spec.
βrj
P2
Z2
subject-object-spec. z ir
global
τj
P3
global
τj
P3
object-spec.
αrj
P4
Z1
ë incl. object-spec. zr
subject-object-spec. z ir
Table 4: Overview over the three possible arguments to supply covariates in BTLLasso
16
Order effects
In principle, the order effect (home advantage in sport events) could also be considered to be a
special case of a subject-object-specific covariate. In the example of Bundesliga matches, the
teams are considered to be the objects and the match days are considered to be the subjects.
Therefore, one can create a subject-object-specific covariate zir representing the home effect,
either using effect coding or dummy coding. With effect coding, one specifies zir 1 if team
ar is the home team on match day i and zis 1 for the away team as , for all other teams
one sets zit 0, t R tr, su. With dummy coding, one specifies zir 1 if team ar is the home
team on match day i and zit 0, t r, otherwise. In both cases, order effects can either
be parameterized with a global parameter δ (global home effect) or with a team-specific
effect δr (team-specific home effect). In BTLLasso, order effects are already implemented
and can be applied using the arguments order.effect and object.order.effect from the
ctrl.BTLLasso() function for global and object-specific order effects, respectively. In the
case of object-specific order effects, the option order.center = TRUE allows for the use of
effect coding instead of dummy coding in the design matrix.
Intercepts
By default, object-specific intercepts are included in the model. To eliminate the intercepts,
the option include.intercepts = FALSE from ctrl.BTLLasso() can be used.
5.2. Specification of penalties
In Section 4.1 several possible penalties were proposed which can be applied to reduce the
complexity of the model and to make the model easier to interpret. All these penalties can be
applied to the specified model using specific options within ctrl.BTLLasso(). In particular,
the user can specify the following arguments:
ˆ penalize.intercepts (de-)activates penalty P1
ˆ penalize.X (de-)activates penalty P2
ˆ penalize.Z2 (de-)activates penalty P3
ˆ penalize.Z1.diffs (de-)activates penalty P4
ˆ penalize.Z1.absolute sets ν1
1 in P4
ˆ penalize.order.effect.diffs (de-)activates penalty P6
ˆ penalize.order.effect.absolute (de-)activates penalty P5 or sets ν2
1 in P6
The arguments can either be set to TRUE or FALSE and, thereby, (de-)activate the respective
penalties.
5.3. Cross-validation
To perform cross-validation of the specified model the function cv.BTLLasso() is used. By
default, this is done as a 10-fold cross-validation (folds = 10) and is parallelized on several
17
cores using the argument cores. One can choose between two different options for the crossvalidation criterion, namely the ranked probability score (RPS) and the deviance. For ordinal
response y P t1, . . . , K u the RPS (Gneiting and Raftery 2007) is recommended which can be
denoted by
RP S py, π̂ pk qq Ķ
pπ̂pkq 1py ¤ kqq2,
k 1
where π pk q represents the cumulative probability π pk q P py ¤ k q. In contrast to the
deviance, the RPS takes advantage of the ordinal structure of the response.
5.4. Bootstrap and confidence intervals
To receive confidence intervals for the parameters a bootstrap method is implemented in the
function boot.BTLLasso(). In each of the B bootstrap iterations the method is applied to
a data set randomly sampled from the original data set, in every iteration the optimal tuning parameter is determined separately by cross-validation to fully account for the additional
variability induced by the cross-validation. Therefore, the bootstrap procedure can be very
time-consuming, although parallelization is possible using the argument cores. In general,
the model generated by cv.BTLLasso() is supplied to boot.BTLLasso(). To speed up computation of the bootstrap, additionally a new (shorter) vector of tuning parameters lambda
can be supplied. Finally, from the bootstrap estimates the requested empirical quantiles are
calculated and confidence intervals can be obtained by using the function ci.BTLLasso().
5.5. Visualizations
In addition to the aforementioned function ci.BTLLasso() for plotting empirical confidence
intervals, two further functions for visualization of the results are provided, namely singlepaths()
and paths(). They can be used to plot the coefficient paths of the estimates produced both by
BTLLasso() and cv.BTLLasso() along the tuning parameter. With singlepaths(), for every
covariate (and for intercepts and order effects), separate plots are generated. These plots are
especially helpful to investigate the clustering the behavior of the penalties along the tuning
parameter and to investigate different covariate effects across objects. In contrast, paths()
generates one path per covariate and, therefore, only generates a single plot containing all
paths. The paths either visualize the penalty terms for the single model components (y.axis
= "penalty") or the L2 norm (y.axis = "L2") of the vector of all respective parameters,
e.g. the L2 norm of all parameters corresponding to one covariate. This type of visualization helps to compare effect sizes or variable importance between different variables or model
components. If a cross-validated model is supplied instead of the regular model, additionally
the optimal model according to cross-validation is highlighted. Both visualization techniques
will be illustrated in the following section. Beside the presented visualization methods, for
objects generated by ci.BTLLasso(), BTLLasso() or cv.BTLLasso() also print-functions
exist to print the most important results of the respective objects into the console.
6. Applications
In the following section, the presented methods are applied to the exemplary data sets of the
German election study and the German Bundesliga as described in Section 3.1.
18
6.1. Election data
In the election data, both subject-specific and subject-object-specific covariates are available.
In contrast to Schauberger and Tutz (2015), where only subject-specific covariates are considered, here both types of covariates are included in the model simultaneously. We choose to
allow for subject-specific effects for the subject-object-specific covariates SocEc, Immigration
and Climate which have to be supplied to cv.BTLLasso() via the Z1 argument. The subjectspecific covariates Age, Gender and Abitur are supplied via X. The following code is used to
fit the optimal cross-validated model to our data:
data(GLES)
Y <- GLES$Y
X <- GLES$X[,c(1,2,5)]
Z1 <- GLES$Z1
lambda <- exp(seq(log(201),log(1),length=100))-1
set.seed(1860)
mcv_GLES <- cv.BTLLasso(Y = Y, X = X, Z1 = Z1, lambda = lambda)
For our model, we wish to penalize both the covariates included in X and Z1. Furthermore, no
order effect is necessary as the order of the parties is random in this application. These model
assumptions match the default settings in ctrl.BTLLasso() and, therefore, no changes have
to be applied to the control argument.
Subsequently, the estimated parameter paths can be plotted using the commands
singlepaths(mcv_GLES, plot.Z1 = FALSE, columns = 3, plot.intercepts = FALSE)
singlepaths(mcv_GLES, plot.X = FALSE, columns = 3, plot.intercepts = FALSE)
separately for all covariates as depicted in Figure 3. Here, columns specifies the number of
plots to place next to each other while commands like plot.Z1 help to specify which model
components should be plotted. In Figure 3, the optimal model according to the 10-fold
cross-validation is highlighted by the red dashed line.
The parameter paths are drawn along the tuning parameter or rather log pλ 1q. On first
sight, a clear difference between the subject-object-specific covariates in Figure 3a and the
subject-specific covariates in Figure 3b becomes obvious. The coefficients in Figure 3a are
not restricted in any way while the coefficients in Figure 3b have to sum up to zero for every
covariate. Therefore, the estimates for subject-object-specific covariates like the attitude
towards climate change can be different from zero but still equal for all parties, which is not
possible (and not sensible) for variables like Gender or Age.
It can be seen, that (according to the red dashed line indicating the optimal model according
to cross-validation) a rather non-sparse model is chosen. Most of the possible covariate effects
differ across parties. For the subject-object-specific variables, all effects are negative. This
result is very intuitive, it simply states that for a higher (self-perceived) distance of voter and
party on a specific issue the attractiveness of the party for the voter decreases. For the socioeconomic dimension, three clusters of parties are distinguished, namely tFDP, SPD, Greensu,
19
SocioEcon
5
4
3
2
1
0
−0.05
Left_Party
5
4
log(λ + 1)
3
2
1
Left_Party
−0.15
−0.44
−0.48
estimates
CDU/CSU
CDU/CSU
−0.52
−0.35
FDP
−0.45
Greens
estimates
SPD
SPD
Greens
FDP
−0.56
−0.25
Left_Party
Immigration
Greens
−0.25
Climate
0
FDP
SPD
CDU/CSU
5
4
log(λ + 1)
3
2
1
0
log(λ + 1)
(a) Subject-object-specific covariates
Gender
0.05
0.04
SPD
FDP
4
3
2
log(λ + 1)
1
0
0.00
estimates
0.02
Greens
Greens
5
Greens
CDU/CSU
FDP
SPD
−0.10
−0.2
Left_Party
CDU/CSU
0.00
0.0
FDP
SPD
estimates
0.1
0.2
CDU/CSU
Abitur
−0.02
0.3
Age
Left_Party
5
4
3
2
1
0
log(λ + 1)
Left_Party
5
4
3
2
1
0
log(λ + 1)
(b) Subject-specific covariates
Figure 1: Parameter paths for subject-object-specific (a) and subject-specific (b) variables of
GLES data. Dashed red lines represent optimal model according to 10-fold cross-validation.
tCDU/CSUu and tLeft Partyu. In that case a cluster means, that for these parties the distance
between party and voter is equally important, it is most important for the attractiveness of the
Left party and less important for the attractiveness of FDP, SPD and Greens. Comparing
the effect sizes, the issue of immigration has the lowest effect. For the climate issue, the
differences between the parties are rather high. Also for the subject-specific covariates rather
many different clusters are found for the single covariates, four clusters for Age and Abitur
and three clusters for Gender. For example, the attractiveness of CDU/CSU is increased with
growing age while the attractiveness of Greens and left party decreases.
Figure 2 was created by the command paths(mcv_GLES, y.axis = "L2") and shows the L2 norms of all covariates along the tuning parameter. The L2 norm measures the size of each
vector which collects all estimates corresponding to one covariate. Therefore, it can be seen
as measure of variable importance as long as the estimates for the scaled covariates are used.
It can be seen that the subject-object-specific covariates representing the different political
dimensions are more important than the subject-specific variables Age, Gender and Abitur.
To conclude the analysis of the election data, we present how to generate empirical bootstrap
confidence intervals. The bootstrap estimation is executed using the following command:
set.seed(1860)
mboot_GLES <- boot.BTLLasso(mcv_GLES, B = 500, cores = 25)
20
SocioEcon
1.0
0.8
Climate
0.6
Immigration
0.4
Age
0.2
Abitur
Gender
0.0
5
4
3
2
1
0
log(λ + 1)
Figure 2: L2 paths for both subject-object-specific and subject-specific variables of GLES
data. Dashed red lines represent optimal model according to 10-fold cross-validation.
Main argument of the function is the object generated from cross-validation. Furthermore,
B specifies the number of bootstrap samples and cores specifies the number of cores to use.
Analogously to the use of singlepaths() before, the function ci.BTLLasso() can be used
to visualize the empirical bootstrap confidence intervals which are shown in Figure 3. In
general, the confidence intervals illustrate the variability of the single estimates and provide
hints if single estimates are distinctly different or not. For example, the confidence intervals
for Age show that for this variable distinct differences exist between the single parties while
this seems not to be the case for Gender.
6.2. Football data
For the data from the German Bundesliga, two different models are fitted for illustration.
In the first model, the subject-object-specific variables Distance and BallPossession are used
to investigate if these variables influence the strength of the different football teams. In the
second model, the influence of the object-specific variable market value is investigated.
The first model in total contains four different model components, namely team-specific home
effects (i.e. subject-specific order effects), intercepts and team-specific effects for both subjectobject-specific variables Distance and BallPossession. The model is computed using the
following code:
data(Buli1516)
Y <- Buli1516$Y
Z1 <- scale(Buli1516$Z1[,1:(2*18)], scale = FALSE)
lambda <- exp(seq(log(31), log(1.01), length = 100)) - 1
ctrl1 <- ctrl.BTLLasso(object.order.effect = TRUE,
name.order = "Home",
penalize.order.effect.diffs = TRUE,
21
Climate
SPD
SPD
●
Left_Party
Left_Party
●
Greens
●
FDP
CDU/CSU
SocioEcon
●
−0.5
−0.3
SPD
●
Greens
●
Greens
FDP
●
FDP
−0.1
●
●
●
CDU/CSU
●
−0.6
●
Left_Party
●
CDU/CSU
●
Immigration
−0.4
−0.2
0.0
●
−0.35
−0.20
−0.05
(a) Subject-object-specific covariates
Age
SPD
●
Left_Party
Greens
Gender
●
FDP
●
SPD
Left_Party
●
Left_Party
FDP
●
CDU/CSU
●
−0.2
SPD
Greens
●
0.0
0.2
Abitur
●
FDP
●
0.05
●
CDU/CSU
●
−0.05 0.00
●
Greens
●
CDU/CSU
●
0.10
●
−0.15
0.00
0.10
(b) Subject-specific covariates
Figure 3: Confidence intervals for subject-object-specific (a) and subject-specific (b) variables
of GLES data according to 500 bootstrap samples.
penalize.order.effect.absolute = FALSE,
order.center = TRUE)
set.seed(1860)
mcv_buli <- cv.BTLLasso(Y = Y, Z1 = Z1, lambda = lambda, control = ctrl1)
The decisive part is the definition of the control argument using ctrl.BTLLasso(). In football
matches, unarguably a home effect is present which is equivalent to the order effect in paired
comparison because the first-named team is the home team. As we do not doubt the general
presence of a home effect, the absolute values of the home effects of the single teams are
not penalized (penalize.order.effect.absolute = FALSE). Only the differences between
the single home effects are penalized (penalize.order.effect.diffs = TRUE) to find out,
whether all teams share the same home effect or if (clusters of) single teams have distinct
home effects. In addition, the covariates in Z1 are centered for a better interpretability of the
intercepts (which are not penalized).
Figure 4 shows the coefficient paths for the four model components along logpλ 1q created
by use of the function singlepaths(). It can be seen that the optimal model in this case is
very sparse. All teams share the same home effect. Furthermore, all teams share the same,
quite strong positive effect of Distance. That means, a strong running performance is strongly
22
Intercepts
DOR
MGB
WOB
S04
MAI
KOE
LEV
BER
HSV
HOF
AUG
BRE
STU
FRA
HAN
ING
DAR
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
log(λ + 1)
log(λ + 1)
Distance
BallPossession
1
−1
0
DOR
WOB
BER
S04
ING
MGB
LEV
MAI
HOF
FRA
AUG
STU
KOE
HAN
BRE
HSV
−2
0.5
S04
BAY
estimates
1.0
1.5
DAR
AUG
MGB
MAI
BRE
ING
DOR
HSV
LEV
STU
HOF
BER
HAN
WOB
KOE
FRA
BAY
0.0
estimates
BAY
−2
0.5
BAY
DOR
FRA
BER
STU
LEV
S04
HOF
MAI
HAN
HSV
BRE
AUG
KOE
DAR
estimates
1.5
ING
−0.5
estimates
WOB
MGB
0 1 2 3 4
Home
DAR
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
log(λ + 1)
log(λ + 1)
Figure 4: Coefficient paths for model with subject-object-specific covariates Distance and
BallPossession and (possibly) team-specific home effects. Red dashed line represents optimal
model according to 10-fold cross-validation.
associated with a better performance of a team. For BallPossession, most teams (except for
Hertha BSC Berlin) share the same negative effect. Therefore, higher values in BallPossession
are associated with rather worse results. Overall, the path plots illustrate the behavior of the
penalty terms. Both for the home effects and for the effects for Distance and BallPossession
the penalty forms clusters of teams with equal effects. With decreasing values of the tuning
parameter, the clusters are decomposed into smaller clusters ending up with single clusters
per team.
The second model contains only three model components, namely the home effects, the intercepts and the effect of the object-specific covariate market value. The model is computed
using the following code:
data(Buli1516)
Y <- Buli1516$Y
Z2 <- Buli1516$Z2
lambda2 <- exp(seq(log(5.5), log(1.0001), length = 100)) - 1
ctrl2 <- ctrl.BTLLasso(object.order.effect = TRUE,
name.order = "Home",
23
penalize.order.effect.diffs = TRUE,
penalize.order.effect.absolute = FALSE,
order.center = TRUE, penalize.intercepts = TRUE)
set.seed(1860)
mcv_buli2 <- cv.BTLLasso(Y = Y, Z2 = Z2, lambda = lambda2, control = ctrl2)
In contrast to the previous model, now the intercepts are also penalized due to the identifiability issues with object-specific covariates already described in Section 3.2. Therefore,
the question answered by the model is whether the market value (which is not penalized) is
sufficient to explain the match outcomes or if, additionally, home effects and intercepts are
necessary for (clusters of) single teams.
Figure 5 shows the coefficient paths for this model created by singlepaths(). Obviously, the
DOR
BAY
−1.5
STU
HSV
HAN
KOE
AUG
0.5
0.0
MGB
LEV
MAI
BER
KOE
ING
S04
HSV
DAR
BRE
AUG
WOB
HOF
FRA
STU
−0.5
BAY
BER
ING
FRA
DOR
HOF
MAI
LEV
S04
BRE
estimates
1.0
MGB
WOB
DAR
1.5
1.0
0.5
log(λ + 1)
0.0
0.35 0.40 0.45 0.50 0.55 0.60
MarketValue
estimates
0.5 1.0 1.5 2.0
Intercepts
−0.5
estimates
Home
HAN
1.5
1.0
0.5
0.0
log(λ + 1)
1.5
1.0
0.5
0.0
log(λ + 1)
Figure 5: Coefficient paths for model with object-specific covariate market value, penalized
intercepts and (possibly) team-specific home effects. Red dashed line represents optimal model
according to 10-fold cross-validation.
market value has a positive effect for the performance of a team. Both for the home effects
and for the intercepts, in the optimal model several clusters occur. For the home effects, seven
different cluster evolve with tMGB, WOBu and tDARu having the best and worst home effect,
respectively. For the intercepts, four clusters are found with Borussia Dortmund forming a
cluster of its own with the highest value.
7. Summary
The R-package BTLLasso presented in this work provides a general framework for the inclusion
of different kinds of covariates in paired comparison models. Covariates can vary over subjects
or objects or both over subjects and objects of paired comparisons. In addition, (possibly
object-specific) order effects can be included. The paired comparisons can be both binary
and ordinal. To obtain sparser models and easier interpretation, suitable penalty terms for
all model components are provided. For all model components with object-specific effects,
the absolute pairwise differences of the parameters can be penalized to force clustering of
24
objects with respect to the corresponding model component. The package contains functions
for cross-validation, bootstrap confidence intervals and for the visualization of results. The
package also provides exemplary data sets to illustrate the possible applications of BTLLasso.
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